Optimality of Gradient-MUSIC for spectral estimation

TL;DR

This work develops Gradient-MUSIC, a grid-free, nonconvex-optimization approach for spectral estimation that simultaneously identifies $s$ frequencies and amplitudes from noisy samples. By coupling a precise coarse-grid initialization on a width-$1/(2m)$ grid with parallel gradient refinements on the continuous domain, Gradient-MUSIC achieves provable, minimax-optimal error bounds in the well-separated regime $m\Delta\ge8\pi$, with frequency error scaling as $\mathcal{O}(\|\boldsymbol\eta\|_p/m^{1+1/p})$ and amplitude error as $\mathcal{O}(\|\boldsymbol\eta\|_p/m^{1/p})$, and analogous high-probability guarantees under stochastic Gaussian noise. The authors provide a comprehensive geometric landscape analysis of the underlying objective, establish minimax lower bounds showing optimality, and demonstrate that oversampling improves recovery accuracy. The framework also accommodates real-amplitude extensions and alternative subspace estimators beyond Toeplitz, illustrating broad applicability. Overall, Gradient-MUSIC offers a faster, grid-free alternative to classical MUSIC with solid theoretical guarantees that match minimax limits in the well-separated setting, including robust performance under stochastic perturbations.

Abstract

We introduce the Gradient-MUSIC algorithm for estimating the unknown frequencies and amplitudes of a nonharmonic signal from noisy time samples. While the classical MUSIC algorithm performs a computationally expensive search over a fine grid, Gradient-MUSIC is significantly more efficient and eliminates the need for discretization over a fine grid by using optimization techniques. It coarsely scans the 1D landscape to find initialization simultaneously for all frequencies followed by parallelizable local refinement via gradient descent. We also analyze its performance when the noise level is sufficiently small and the signal frequencies are separated by at least $8π/m$, where $π/m$ is the standard resolution of this problem. Even though the 1D landscape is nonconvex, we prove a global convergence result for Gradient-MUSIC: coarse scanning provably finds suitable initialization and gradient descent converges at a linear rate. In addition to convergence results, we also upper bound the error between the true signal frequencies and amplitudes with those found by Gradient-MUSIC. For example, if the noise has $\ell^\infty$ norm at most $\varepsilon$, then the frequencies and amplitudes are recovered up to error at most $C\varepsilon/m$ and $C\varepsilon$ respectively, which are minimax optimal in $m$ and $\varepsilon$. Our theory can also handle stochastic noise with performance guarantees under nonstationary independent Gaussian noise. Our main approach is a comprehensive geometric analysis of the landscape, a perspective that has not been explored before.

Figures (4)

• Figure 1: Plot of the landscape function $q$ generated from ${\boldsymbol x}=\{2\pi(2/m),\, 2\pi(10/m),\, \pi\}$ and the landscape function $\widetilde{q}$ associated with $\widetilde{{\boldsymbol U}}$.
• Figure 2: Plot of $\widetilde{q}$ and the set of $(u,\widetilde{q}(u))$, where $u$ are elements of a uniform grid of width $2\pi/(8m)$. The accepted and rejected $(u,q(u))$ are shown in green and red, respectively.
• Figure 3: Diagram of the map $\mathbb{F}^{m\times s}\subseteq\mathbb{U}^{m\times}$ to $\mathcal{F}_{m,s}\subseteq \mathcal{U}_{m,s}$. Arbitrary ${\boldsymbol U}\in \mathbb{F}^{m\times s}$ and $\widetilde{{\boldsymbol U}}\in \mathbb{U}^{m\times s}$ are mapped to their associated landscape functions $q_{{\boldsymbol U}}\in \mathcal{F}_{m,s}$ and $q_{\widetilde{{\boldsymbol U}}}\in \mathcal{U}_{m,s}$.
• Figure 4: Gradient-MUSIC frequency and amplitude errors for nonstationary independent Guassian noise.

Theorems & Definitions (68)

• Definition 2.1: Spectral Estimation Problem
• Remark 2.2
• Remark 2.3
• Proposition 2.1: Aubel-Bölcskei aubel2019vandermonde
• Remark 2.4
• Definition 2.5: Steering vector
• Definition 2.6: Landscape function
• Proposition 2.2
• Definition 2.7: Toeplitz Estimator
• Definition 3.1: Accepted elements